To Determine Young's Modulus of Elasticity of the Material of a Given Wire
1. Aim
To determine the Young's modulus of elasticity of the material of a given wire using Searle's apparatus.
2. Apparatus Used
- Searle's apparatus with experimental and reference wires
- A set of weights/slotted masses (50g, 100g, 200g, etc.)
- Traveling microscope with vernier scale
- Meter scale
- Screw gauge (for measuring diameter of wire)
- Weight hanger
- Spirit level
- Vernier calipers (optional)
3. Diagram
Fig. 1: Schematic diagram of Searle's apparatus for determining Young's modulus
4. Theory
Young's modulus is a mechanical property that measures the stiffness of a solid material. It quantifies the relationship between stress (force per unit area) and strain (proportional deformation) in a material undergoing tension or compression in only one direction.
When a wire is subjected to a tensile force, it undergoes elongation. According to Hooke's law, within the elastic limit, the stress is directly proportional to the strain:
$\text{Stress} \propto \text{Strain}$
$\text{Stress} = E \times \text{Strain}$
Where E is the Young's modulus of elasticity, which is a characteristic property of the material.
In this experiment, we use Searle's apparatus which consists of two identical wires: an experimental wire and a reference wire. The experimental wire is stretched by adding weights, causing it to elongate. The elongation is measured using a vernier microscope or scale. By analyzing the relationship between the applied load and the extension produced, we can determine the Young's modulus of the material.
5. Formula
Young's modulus (E) is calculated using the following formula:
$$E = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{l/L}$$
$$E = \frac{F \times L}{A \times l}$$
$$E = \frac{mg \times L}{\pi r^2 \times l}$$
Where:
- E = Young's modulus of elasticity (N/m² or Pascal)
- F = Applied force (N)
- A = Cross-sectional area of the wire (m²)
- L = Original length of the wire (m)
- l = Extension produced in the wire (m)
- m = Mass added (kg)
- g = Acceleration due to gravity (9.8 m/s²)
- r = Radius of the wire (m)
The cross-sectional area of the wire is calculated as:
$$A = \pi r^2 = \pi \left(\frac{d}{2}\right)^2$$
Where d is the diameter of the wire.
6. Procedure
- Measure the diameter of the experimental wire at different positions using a screw gauge and calculate the average value.
- Calculate the radius and cross-sectional area of the wire.
- Measure the original length (L) of the wire between the two fixed points.
- Ensure that the apparatus is properly leveled using the spirit level.
- Set up the traveling microscope to observe the reference mark on the experimental wire.
- Record the initial reading of the microscope when no load is applied.
- Add a weight (e.g., 50g) to the weight hanger attached to the experimental wire.
- Wait for the system to stabilize (approximately 2-3 minutes).
- Record the new position of the reference mark using the microscope.
- Calculate the extension (l) as the difference between the final and initial readings.
- Repeat steps 7-10 for different loads, incrementing by fixed amounts (e.g., 50g or 100g).
- Gradually remove the weights and check if the wire returns to its original position (to verify it's within elastic limit).
- Plot a graph of load (F) versus extension (l).
- Calculate Young's modulus using the slope of the graph or the formula provided.
7. Observation Table
Measurement of Wire Diameter:
Least count of screw gauge = ____ mm
Zero error of screw gauge = ____ mm
| Observation | Main Scale Reading (mm) | Circular Scale Reading | Total Reading (mm) |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| Mean diameter (d) | ____ mm | ||
Length of the wire (L) = ____ m
Measurement of Extension:
Least count of microscope = ____ mm
| S.No. | Load, m (kg) | Force, F = mg (N) | Initial Reading (mm) | Final Reading (mm) | Extension, l (mm) |
|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | ||
| 2 | 0.05 | 0.49 | |||
| 3 | 0.10 | 0.98 | |||
| 4 | 0.15 | 1.47 | |||
| 5 | 0.20 | 1.96 | |||
| 6 | 0.25 | 2.45 | |||
| 7 | 0.30 | 2.94 |
8. Calculations
From the collected data:
-
Mean diameter of the wire (d) = ____ m
Radius of the wire (r) = d/2 = ____ m
-
Cross-sectional area of the wire:
$$A = \pi r^2 = \pi \times (\_\_\_\_ \text{ m})^2 = \_\_\_\_ \text{ m}^2$$
-
Length of the wire (L) = ____ m
-
For each observation, calculate Young's modulus:
$$E = \frac{F \times L}{A \times l} = \frac{mg \times L}{\pi r^2 \times l}$$
-
Example calculation for one observation (e.g., for load = 0.25 kg):
$$E = \frac{0.25 \times 9.8 \times L}{\pi \times r^2 \times l} = \_\_\_\_ \text{ N/m}^2$$
-
Mean value of Young's modulus:
$$E_{mean} = \frac{E_1 + E_2 + E_3 + \ldots + E_n}{n} = \_\_\_\_ \text{ N/m}^2 = \_\_\_\_ \times 10^{11} \text{ Pa}$$
9. Result
The Young's modulus of elasticity of the material of the given wire is found to be:
$$E = \_\_\_\_ \times 10^{11} \text{ N/m}^2 \text{ or } \_\_\_\_ \times 10^{11} \text{ Pa}$$
The expected value for common materials:
- Steel: 2.0 × 10¹¹ Pa
- Copper: 1.1 × 10¹¹ Pa
- Aluminum: 0.7 × 10¹¹ Pa
- Brass: 0.9 × 10¹¹ Pa
The experimental value is close to the standard value for _______ (name of material), with a percentage error of _____%. This indicates that the experiment was conducted with reasonable accuracy.
10. Precautions
- Ensure that the wire is straight and free from kinks before starting the experiment.
- The apparatus should be placed on a stable surface and properly leveled.
- Load should be applied gradually to avoid sudden jerks or oscillations.
- Allow sufficient time after applying each load for the system to stabilize before taking readings.
- Ensure that the load does not exceed the elastic limit of the wire.
- The wire should be free from rust and impurities.
- Avoid parallax error while taking readings from the traveling microscope or scale.
- Measure the diameter of the wire at different positions to account for any variations.
- Ensure that the reference wire is properly aligned and does not interfere with the experimental wire.
- Check that the zero error of all measuring instruments is noted and accounted for in calculations.
- The experiment should be conducted at a constant temperature to avoid thermal expansion effects.
11. Sources of Error
- The wire may not be perfectly uniform in cross-section throughout its length.
- Initial straightening of the wire may be misinterpreted as extension due to the applied load.
- Friction at pulleys and supports can affect the actual load applied to the wire.
- Temperature variations during the experiment can cause thermal expansion or contraction of the wire.
- Errors in measuring the diameter of the wire directly affect the calculation of cross-sectional area.
- Errors in reading the microscope or scale can lead to inaccurate extension measurements.
- The wire may have undergone some permanent deformation if loaded beyond its elastic limit in previous tests.
- Air currents or vibrations in the laboratory can cause slight oscillations in the system.
- The value of acceleration due to gravity (g) may vary slightly from the standard value of 9.8 m/s².
- Backlash error in the microscope adjustment mechanism.
12. Viva Voice Questions
Q1: What is Young's modulus and what does it physically represent?
A1: Young's modulus is a mechanical property that measures the stiffness of a solid material. It represents the ratio of stress (force per unit area) to strain (proportional deformation) within the elastic limit of a material. Physically, it indicates how much a material will deform (stretch or compress) when subjected to a tensile or compressive force.
Q2: Why is Young's modulus important in engineering applications?
A2: Young's modulus is crucial in engineering design because it helps predict how materials will behave under load. It's used to calculate deflections, ensure structural stability, select appropriate materials for specific applications, and design components that can withstand expected stresses without excessive deformation.
Q3: What is Hooke's law and how is it related to Young's modulus?
A3: Hooke's law states that, within the elastic limit, the strain produced in a body is directly proportional to the stress applied to it. Young's modulus is the constant of proportionality in Hooke's law when applied to tensile or compressive stress-strain relationships.
Q4: Why do we use a reference wire in Searle's apparatus?
A4: The reference wire helps compensate for any temperature changes that might occur during the experiment. Since both wires experience the same temperature variations, any thermal expansion or contraction effects are nullified when measuring the extension of the experimental wire relative to the reference wire.
Q5: How would you determine if the wire has exceeded its elastic limit during the experiment?
A5: If the wire has exceeded its elastic limit, it will not return to its original length when the load is removed. By gradually removing the weights and checking if the wire returns to its initial position, we can verify if it's still within the elastic region.
Q6: Why is the cross-sectional area of the wire squared in the denominator of the Young's modulus formula?
A6: The Young's modulus formula contains the cross-sectional area in the denominator because stress is calculated as force divided by area. The formula for Young's modulus is E = (F/A)/(ΔL/L), where F/A represents stress and ΔL/L represents strain.
Q7: How does temperature affect Young's modulus?
A7: Generally, Young's modulus decreases with increasing temperature for most materials. This happens because higher temperatures increase atomic vibrations and weaken interatomic bonds, making the material less stiff and more easily deformed.
Q8: Compare the Young's modulus of metals, polymers, and ceramics.
A8: Metals typically have Young's modulus values in the range of 45-210 GPa, with steel being among the stiffest. Ceramics often have even higher values, ranging from 70-300+ GPa, making them very rigid but brittle. Polymers have much lower Young's modulus values, typically between 0.5-10 GPa, making them more flexible than metals and ceramics.
Q9: What is the difference between stress, strain, and Young's modulus?
A9: Stress is the force applied per unit area (N/m² or Pa). Strain is the proportional deformation that occurs in response to stress (dimensionless). Young's modulus is the ratio of stress to strain, representing the stiffness of the material (also in Pa).
Q10: Why is it important to measure the diameter of the wire at multiple positions?
A10: Wires may not have perfectly uniform diameter throughout their length due to manufacturing variations. By measuring at multiple positions and taking an average, we get a more accurate representation of the cross-sectional area, which is critical for calculating Young's modulus correctly.